Algorithms and Data Structures for Database Systems · R121 R122 R211 R212 R1 R2 R1 R2 R11 R12 R13...

Algorithms and Data Structures for Database Systems

Jürgen Treml2005-06-08

Overview

1. Introduction, Motivation2. R-tree Index Structure – Overview3. Algorithms on R-trees

SearchingInsert / DeleteNode SplittingUpdates & other Operations

4. Performance, Benchmarks5. R-tree Modifications6. Conclusion

Overview – Where are we?




Introduction, Motivation

Importance of effectively storing and indexing spatial data (CAD, VR, Image Processing, Cartography, …)One-dimensional indexes not suitableStrong limitations with most spatial structures (e.g. point data only, not dynamic, performance with paged memory, …)

Introduction, Motivation

Antonin Guttman, 1984“A Dynamic Index Structure For Spatial Searching”Based on B-trees

Region-trees (R-trees)

R-tree Index StructureData objects are re-

presented by their MBRs

R-tree Index Structure

Formal DescriptionStructure consisting of (regular) nodes containing tuples ),( CPI n

At the lowest level: leaf-nodes with tuples),( TIDIn

MBR (minimum bounding rectangle)),...,,( 10 nn IIII =


Important ParametersMaximum number of elements per node

MMinimum number of elements per node

2Mm ≤

Height of an R-tree⎡ ⎤ 1log −Nm


Important RestrictionsAll leaf-nodes are on the same levelRoot has at least two children (unless it is a leaf)

Algorithms on R-trees: Search

Similar to B-tree searchQuite easy & straight forward(Traverse the whole tree starting at the root node)

No guarantee on good worst-case performance!(Possible overlapping of rectangles of entries within a single node!)


Short Description:For all entries in a non-leaf node:Check if overlapping

If yes: check node pointed to by this entryIf node is a leaf-node:Check all entries if overlapping the search object

If yes: entry is a qualifying record!


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Algorithms on R-trees: Insert

Again: Similar to corresponding B-tree operationBasically consist of 3 parts:

1. CHOOSELEAF(Find place to insert new object)

2. INSERT(Insert the new object)

3. ADJUSTTREE(Adjusting preceding nodes)


Short Description:CHOOSELEAF:Start with root Run through all nodes:Find the one which would have to be least enlarged to include given object!INSERT:Check if room for another entry

Insert new entry directly OR after calling SPLITNODE (in case of no room)


Short Description:CHOOSELEAF (…)INSERT (…)ADJUSTTREE:Ascend from node with new entry:Adjust all MBRs!In case of node split:Add new entry to parent node(If no room in parent node, invoke SPLITNODE again)Propagate upwards till root is reached!


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Find node to insert the new object into:

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Leaf is full! Overflow!

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Algorithms on R-trees: InsertR11

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Algorithms on R-trees: Delete

NOT similar to B-tree DELETE(Treatment of underflows)B-tree:Merge under-full node with “neighbor”R-tree:Delete under-full node and re-insertWhy?Re-use of INSERT routineIncrementally refines spatial structure

Algorithms on R-trees: Delete

“Very” Short DescriptionFINDLEAF:Locate the leaf that contains object to be deletedDELETE:Delete entry from nodeCONDENSETREE:Delete and re-insert node if under-full (and in the following all resulting under-full nodes)

Algorithms on R-trees: DeleteR1 R2

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Algorithms on R-trees: Splitting

Splitting of nodes necessary after underflow or overflow (as a result of a delete or insert operation)Ultimate goal: Minimize the resulting node’s MBRsSecondary aim: Do it fast! ;-)3 Implementations by Guttmann:Exhaustive, Quadratic-cost, Linear-cost


Minimal resulting MBRs:

Bad split Good split

Exhaustive Approach:Simply try all possible split-ups!


Quadratic-cost Algorithm:Pick the 2 out of M entries that would consume the most space if put together.

Put one in each groupFor all remaining entries: Pick the one that would make the biggest difference in area when put to one of the two groups

Add it to the one with the least differenceFinished when all entries are put in either group!Quadratic in M, linear in the number of dimensions


Linear-cost Implementation:Basically the same as quadratic-cost algorithmDifferences:First pair is picked by finding the two rectangles with the greatest normalized separation in any dimension.Remaining pairs are selected randomly.Linear in M as well as in the number of dimensions

Algorithms on R-trees: Other Ops

Modified search algorithmKey search / search for specific entries

Range deletion

R-trees are quite as well extensible and efficient as B-trees for the matter of algorithms

Performance of R-trees

Why do Benchmarking?Proof practicality of the structureFind suitable values for m and MTest various node-splitting algorithms

Testing environment:C-implementation running under UNIXLayout of a RISC-II chip with 1057 rectanglesDifferent page sizes (128 – 2048 bytes)Various values for M (6 – 102)

Performance of R-treesLinear INSERT fastestLinear INSERT quite insensitive to M and mQuadratic INSERT depends on M as well as mDELETE extremely sensitive to m

Performance of R-treesSame page hit / miss performance for linear and quadratic splitSlight advantage for exhaustive version ;-)Space usage strongly depending on m

Performance of R-treesQuadratic-cost splitting with jumps at INSERT operations for increasing amount of dataLinear INSERT extremely constantR-tree structure very effective in directing search to small subtrees

R-tree Modifications

Why?Improve performanceOptimize space usage

Different forms of R-trees:R+-trees: Reduce overlapping MBRs Increase search performanceR*-trees: Take overlapping and circumference in to consideration when performing INSERT or DELETE

R-tree Modifications

TV-trees: Allow more complex objects than MBRsX-trees: Avoid / reduce overlapping by dynamically adjusting node capacitiesQR-trees: Hybrid concept, combining R-trees with quad trees

Conclusions

Reasons for R-trees:Importance of being able to store and effectively search spatial dataExisting structures with many limitations

Basic Idea:Structure similar to B-treeRepresent objects by their MBRAllow overlapping

ConclusionsAlgorithms:

Search and insert similar to B-tree operationsDelete performing re-insert instead of merge for under-full nodes (incrementally refines spatial structure)Node splitting: Benchmarks show that linear version performs quite as well as quadratic-cost implementation

Modifications:Structure is easy to adapt for special applications and their needsPerformance advantages are usually paid for with price of a structure that gets harder to maintain

R-trees are the spatial correspondent of B-trees!

Algorithms and Data Structures for Database Systems · R121 R122 R211 R212 R1 R2 R1 R2 R11 R12 R13...

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